Gruber P. Convex and Discrete Geometry

XII Contents
20 Linear Optimization . ....................................... 335
20.1 Preliminaries and Duality . ............................ 336
20.2 TheSimplexAlgorithm............................... 339
20.3 The Ellipsoid Algorithm . . ............................ 343
20.4 Lattice Polyhedra and Totally Dual Integral Systems ....... 345
20.5 HilbertBasesandTotallyDualIntegralSystems .......... 348
Geometry of Numbers and Aspects of Discrete Geometry ............... 353
21 Lattices................................................... 355
21.1 Basic Concepts and Properties and a Linear Diophantine
Equation ........................................... 356
21.2 CharacterizationofLattices............................ 359
21.3 Sub-Lattices . ....................................... 361
21.4 PolarLattices ....................................... 365
22 Minkowski’s First Fundamental Theorem . ..................... 366
22.1 The First Fundamental Theorem . . . ..................... 366
22.2 Diophantine Approximation and Discriminants
of Polynomials . . . . . . ................................ 370
23 Successive Minima . . ....................................... 375
23.1 Successive Minima and Minkowski’s Second Fundamental
Theorem ........................................... 376
23.2 Jarník’s Transference Theorem and a Theorem of Perron
andKhintchine ...................................... 380
24 The Minkowski–Hlawka Theorem ............................ 385
24.1 The Minkowski–Hlawka Theorem . ..................... 385
24.2 Siegel’s Mean Value Theorem and the Variance Theorem
of Rogers–Schmidt . . . ................................ 388
25 Mahler’s Selection Theorem . ................................ 391
25.1 Topology on the Space of Lattices . ..................... 391
25.2 Mahler’s Selection Theorem . . . . . . ..................... 392
26 The Torus Group E d /L ...................................... 395
26.1 Definitions and Simple Properties of E d /L ............... 395
26.2 The Sum Theorem of Macbeath–Kneser . . . .............. 398
26.3 Kneser’s Transference Theorem . . . ..................... 403
27 Special Problems in the Geometry of Numbers . . . . .............. 404
27.1 The Product of Inhomogeneous Linear Forms and DOTU
Matrices............................................ 405
27.2 Mordell’s Inverse Problem and the Epstein Zeta-Function . . 408
27.3 Lattice Points in Large Convex Bodies . . . . .............. 410
28 Basis Reduction and Polynomial Algorithms . . . . . . .............. 411
28.1 LLL-Basis Reduction . ................................ 411
28.2 Diophantine Approximation, the Shortest and the Nearest
Lattice Vector Problem . . . ............................ 417